Hitherto, there has been known a control device for a transmission that is capable of establishing a one-way transmission speed at which rotational driving force is transferred from an input shaft to an output shaft but the rotational driving force is not transferred from the output shaft to the input shaft (refer to Patent Document 1, for example). The transmission establishes the one-way transmission speed by engaging both a first engagement element among a plurality of engagement elements and a one-way clutch. In a case in which a shift speed is established by engaging both the first engagement element and an engagement element other than the one-way clutch when stopping (idle stop) an operation of an engine due to an establishment of an idle stop condition including a case in which the vehicle becomes stationary and a case in which the output of the engine decreases in a so-called coast state, the aforementioned control device disengages the engagement element other than the one-way clutch. Thereby, the one-way transmission speed is established by the one-way clutch being engaged while the engagement of the first engagement element is maintained. Therefore, it is possible to improve responsiveness of driving force transmission at a time of re-start of the engine while avoiding drag of the engine during the engine being stopped. In the transmission described in Patent Document 1, the hydraulic pressure from an electrically powered pump is supplied to the first engagement element while the operation of a mechanical oil pump is stopped due to the operation stop of the engine.